Note that if \(\varphi \in \Phi _w(\Omega )\)  satisfies (aDec)\(_q\)  for \(1\leqslant q<\infty \) , then, by [1]} (since \(\varphi \)  satisfies (aInc)\(_1\)  by definition) we have
\(\Vert u\Vert _\varphi \lesssim \max \lbrace \varrho _\varphi (u),\varrho _\varphi (u)^{\frac{1}{q}}\rbrace .\) 
